Change the export stl file code from Exported from Blender-x.x.x to object name
Slippery Slope.
Would investigate doing this in a post pro way, Using something like sed   or python, to run thru stl files and automate the bothersome edit, rather than updating the addon code.
That said, a quick edit to change the header when writing to ASCII
stl_utils.py
def _ascii_write(filepath, faces):
    from mathutils.geometry import normal

    with open(filepath, 'w') as data:
        fw = data.write
        header = _header_version()
        fw('solid %s\n' % header)

        for face in faces:
            # calculate face normal
            fw('facet normal %f %f %f\nouter loop\n' % normal(*face)[:])
            for vert in face:
                fw('vertex %f %f %f\n' % vert[:])
            fw('endloop\nendfacet\n')

        fw('endsolid %s\n' % header)

change the code (circa line 205) from
header = _header_version()

to
header = filepath

or to hack it just to use the filepath stem &quot;/path/foo/bar/stem.extension&quot;
from pathlib import Path
header = Path(filepath).stem

or to hack it just to use the filname (with extension) &quot;/path/foo/bar/filename&quot;
from pathlib import Path
header = Path(filepath).name